This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present invention. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the present invention. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
Seismic inversion is a process of extracting information about the subsurface from data measured at the surface of the Earth during a seismic acquisition survey. In a typical seismic survey, seismic waves are generated by a source 101 positioned at a desired location. As the source generated waves propagate through the subsurface, some of the energy reflects from subsurface interfaces 105, 107, and 109 and travels back to the surface 111, where it is recorded by the receivers 103. The seismic waves 113 and 115 that have been reflected in the subsurface only once before reaching the recording devices are called primary reflections. In contrast, multiple reflections 117 are the seismic waves that have reflected multiple times along their travel path back to the surface (dashed lines in FIG. 1). Surface-related multiple reflections are the waves that have reflected multiple times and incorporate the surface of the Earth or the water surface (more generally, this is an interface with air, which may be a water-air interface in the case of marine data or land-air interface in the case of land data) in their travel path before being recorded.
During seismic, electromagnetic, or a similar survey of a subterranean region, geophysical data are acquired typically by positioning a source at a chosen shot location, and measuring seismic, electromagnetic, or another type of back-scattered energy generated by the source using receivers placed at selected locations. The measured reflections are referred to as a single “shot record”. Many shot records are measured during a survey by moving the source and receivers to different locations and repeating the aforementioned process. The survey can then be used to perform Inversion, e.g., Full Waveform/Wavefield Inversion (FWI) in the case of seismic data, which uses information contained in the shot records to determine physical properties of the subterranean region (e.g., speed of sound in the medium, density distribution, resistivity, etc. . . . ). Inversion is an iterative process, each iteration comprising the steps of forward modeling to create simulated (model) data and objective function computation to measure the similarity between simulated and field data. Physical properties of the subsurface are adjusted at each iteration to ensure progressively better agreement between simulated and field data.
FWI is a seismic method capable of utilizing the full seismic record, including the seismic events that are treated as “noise” by standard inversion algorithms. The goal of FWI is to build a realistic subsurface model by minimizing the misfit between the recorded seismic data and synthetic (or modeled) data obtained via numerical simulation.
FWI is a computer-implemented geophysical method that is used to invert for subsurface properties such as velocity or acoustic impedance. The crux of any FWI algorithm can be described as follows: using a starting subsurface physical property model, synthetic seismic data are generated, i.e. modeled or simulated, by solving the wave equation using a numerical scheme (e.g., finite-difference, finite-element etc.). The term velocity model or physical property model as used herein refers to an array of numbers, typically a 3-D array, where each number, which may be called a model parameter, is a value of velocity or another physical property in a cell, where a subsurface region has been conceptually divided into discrete cells for computational purposes. The synthetic seismic data are compared with the field seismic data and using the difference between the two, an error or objective function is calculated. Using the objective function, a modified subsurface model is generated which is used to simulate a new set of synthetic seismic data. This new set of synthetic seismic data is compared with the field data to generate a new objective function. This process is repeated until the objective function is satisfactorily minimized and the final subsurface model is generated. A global or local optimization method is used to minimize the objective function and to update the subsurface model.
Because of the strong nonlinearity inherent in the FWI method, the results often depend on the quality of the starting velocity model. Choosing a poor starting model leads to the well-known phenomenon of “cycle-skipping” (when the traveltime difference between events simulated numerically in the computer and those acquired in the field exceeds half the period corresponding to the dominant frequency of the data) and results in the optimization process converging to an undesirable local minimum. A significant contributing factor is the accumulation of error as events are simulated using an inaccurate model of the subsurface. This error is especially large for the so-called multiple reflections, which travel down to the subsurface and then back to the acquisition surface, where they get reflected down and repeatedly propagate through the subsurface region, being recorded each time they reach the surface where receivers are located. As an example, if due to model inaccuracies a primary event was simulated with a traveltime error of Δt, then the corresponding first-order multiple would be simulated with an error of 2Δt, the second-order multiple would have an error of 3Δt, etc. . . .
Unfortunately, correct modeling of multiples is challenging because even small inaccuracies in the background velocity model or mispositioning of the multiple-generating horizons can lead to significant accumulation of traveltime errors as multiples repeatedly traverse the subsurface.
Ideally, one would like to turn this sensitivity of multiples to the inaccuracy of the model into an advantage by converting the traveltime and amplitude errors into a source of information about the subsurface to be used in FWI.
One way to overcome the issue of error accumulation is to model multiples differently. Instead of performing numerical simulation using a point source and a free surface boundary condition, one can inject recorded field data traces as sources at receiver locations (Amundsen and Robertsson, 2014) and let the resulting wavefield propagate through the subsurface one more time. This spatially distributed source derived from field data will be called “areal” source hereafter. If an absorbing boundary condition is used instead of the free surface one and the direct arrival (from the source to receivers) is muted out, then primaries in the recorded field data will be converted into first-order multiples, first-order multiples will be converted into second-order multiples, etc. . . . Also note that, if the direct arrival is not muted out, it will be converted into primaries (perhaps with a phase error, depending on implementation).
Because field data does not have any numerical traveltime errors embedded in it and is propagated through the subsurface just once thanks to the absorbing boundary condition, no accumulation of errors occurs and, using the notation introduced in the first paragraph above, the traveltime error for all orders of multiples will be bounded by Δt. This approach is well-known and has been used most recently in the FWI context by Zhang et al. (2013), although they did not point out this particular advantage of the method. Moreover, their proposed inversion method required extracting multiples from the field data (e.g., using SRME) so that FWI could be performed by comparing modeled multiples with those extracted from the field data.
A better approach would be to avoid the need to estimate multiples separately and perform FWI by using all recorded events, both primaries and multiples. This, in turn, necessitates modeling primaries and multiples at the same time. In a recent publication, Tu and Herrmann (2014) proposed injecting a point source at the physical source location together with an areal source (and an absorbing boundary condition) to model primaries and multiples at the same time for the purposes of imaging (also known as migration or linearized inversion). They propagate the two source functions through a smooth medium, as is common in imaging, and then correlate the resulting wavefield with the back-propagated data in order to obtain the subsurface reflectivity. As their paper points out, a disadvantage of this approach is the generation of “cross-talk” when multiples of different orders happen to correlate with each other, thus producing “fake” undesirable events.